DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

COLLEGE OF ENGINEERING AND TECHNOLOGY

OLD DOMINION UNIVERSITY

NORFOLK, VIRGINIA 23529

GUIDANCE AND CONTROL STRATEGIES

FOR AEROSPACE VEHICLES

By

Joseph L. Hibey, Principal Investigator

and

Desineni S. Naidu, Co-Principal Investigator

Final Report

For the period ended July 31, 1990

Prepared for

National AeronaUtics and Space Administration

Langley Research Center

Hampton, Virginia 23665

Under

Research Grant NAG-I-736

Dr. Daniel D. Moerder, Technical Monitor

GCD-Spacecraft Controls Branch

Submitted by the

Old Dominion University Research Foundation

P.O. Box 6369

Norfolk, Virginia 23508-0369

August 1990
GUIDANCE AND CONTROL STRATEGIES FOR AEROSPACE VEHICLES

(NASA Langley Grant NAG1-736)

PART I

This final report under NASA Langley grant NAGI-736 consists of two

parts. Part I consists of the summary of the earlier work whose reports have

been submitted at various times during the period (I/I/87 to 12/31/1989). Part

I! describes in detail the research work done during the period I/I/1990 to

7/31/1990.
TABLE OF CONTENTS

Paze

I. INTRODUCTION ............................................... 3

II. SUMMARY/ABSTRACT ........................................... 7

III. LIST OF PUBLICATIONS ...................................... 21

2
I. INTRODUCTION

The first part of the report concerns broadly the summary of the work

done in the areas of singular perturbations and time scales (SPATS),

aerobraking technology, guidance and aerocruise.

(i) SPATS

The dynamics of many control systems is described by high-order

differential equations containing parameters such as small time constants,

masses, moments of inertia, inductances, and capacitances. The presence of

these "parasitic" parameters is often the source for the increased order and

the "stiffness" of the system. The "curse" of the dimensionality coupled with

Lhe stiffness poses formidable computational complexitles for the analysis and

control of such large systems. Singularly perturbed systems are those whose

order is reduced when the parasitic parameter is neglected. The methodology of

singular perturbations and time scales (SPATS) is a "boon" to control

engineers in tackling these large scale systems. As such it is very desirable

to formulate many control problems to fit into the framework of the

mathematical theory of SPATS.

The methodology of SPATS has an impressive record of applications in a

wide spectrum of fields including flight mechanics and trajectory

optimization. The aerospace problems involve, in general, the solution of

nonlinear differential equations by resorting to numerical integration.

Analytical solutions are important in providing a general understanding of the

structure of solutions and a better foundation for the solution of guidance

problems. With this in view, attempts have been made to obtain approximate

analytical solutions for the atmospheric entry problem using asymptotic

methods such as the method of matched asymptotic expansions, singular

perturbation method, and multiple scale method.

In this report, using the theory of SPATS, the various types of

aerospace-related problems investigated were digital flight control systems

[I-3, 6-8, I0, 14, 16] and atmospheric entry problems [9-11, 18, 21, 22].

" The numbers in brackets indicate the item under List of Publications.

3
(ii) Aerobraking Technology

The specification spectrum for the proposed Space Transportation System

(STS) places heavy emphasis on the development of reusable avionics subsystems

having special features such as vehicle evaluation and reduction of ground

support for mission planning, contingency response and verification and

validation. According to the report of the National Commission on Space,

PIONEERING THE SPACE FRONTIER, the concept of aerobraking for orbit transfer

has been recognized as one of the critical technologies and recommended for

demonstration projects in building the necessary technology base for

pioneering the space frontier. In space transportation systems, the

aerobraking, defined as the deceleration resulting from the effects of

atmospheric drag upon a vehicle during orbital operations, opens new mission

opportunities, especially with regard to the initiation of a permanent space

station.

The main function of space transportation system is to deliver payloads

from Earth to various locations in space. Until now, this function has been

performed by various rockets, the space shuttle, and expendable upper stages

using solid or liquid propellants. In particular, considering the economic

benefits and reusability, an orbital transfer vehlcle (OTV) is proposed for

transporting payloads between low Earth orbit (LEO) and high Earth orbit

(HEO). The two basic operating modes contemplated for OTV are a ground-based

OTV which returns to Earth after each mission and a space-based OTV which

operates out of an orbiting hanger located at the proposed Space Station

Freedom.

In a typical mission, a space-based OTV, which is initially at the space

station orbit (SSO), is required to transfer a payload to geosynchronous Earth

orbit {CEO), pick up another payload, say a faulty satellite, and return to

mate with the orbiting hanger at SSO for refurbishment and redeployment of the

payload. The OTV on its return journey from CEO to SSO needs to dissipate some

of its orbital energy. This can be accomplished by using an entirely

propulsive (Hohmann) transfer in space only or a combination of propulsive

transfer in space and aerobraking maneuver in the atmosphere. ]it has been

established that a significant fuel savings and hence increased payload

capabilities can be achieved with propulsive and aerobraking {or aeroassisted)

4
maneuvers instead of all-propulsive maneuvers. The word "aeroassisted" is

often used to convey that the atmosphere is used to achieve the desired

deceleration. This leads to an aeroassisted orbital transfer vehicle (AOTV),

which on its return leg of the mission, dips into the Earth's atmosphere,

utilizes atmospheric drag to reduce the orbital velocity and employs lift and

bank angle modulations to achieve a desired orbital incllnatlon. Basically,

the AOTV performs a synergetic maneuver, employing a hybrid combination of

propulsive maneuver in space and aerodynamic maneuver in the atmosphere.

Broadly speaking, the two kinds of orbital transfer are coplanar orbital

transfer and noncoplanar orbital transfer (or orbital transfer with plane

change).

In this report, using algorithms based on industry standard program to

optimize simulated trajectories (POST), and multiple shooting method,

investigations have been carried out to generate fuel-optimal trajectories for

coplanar orbital transfer [4, 12, 17, 25], and noncoplanar orbital transfer

[13, 15, 19, 20, 29] arising in aerobraking technology.

(iii) Guidance

An optimal trajectory is computed for a given nonlinear dynamical system

with a fixed set of conditions. However, variation of the initial and final

conditions, plant parameters would alter the optimal trajectory. It is

computationally tedious and expensive to repeat the whole optimization

procedure for every changed condition and obtain a new optimal trajectory. In

such a situation, an alternative is to linearize the original system and

generate an optimal trajectory in the neighborhood of the original optimal

trajectory, involving considerably less computational effort.

Cuidance is the determination of a strategy for following a nominal

flight in the presence of off-nominal conditions, wind disturbances, and

navigation uncertainties. In a typical guidance scheme, the final steering

command is generated as the sum of two components, an open-loop actuating

(control) signal yielding the desired vehicle trajectory in the absence of

external disturbances, and a linear feedback regulating signal which reduces

the system sensitivity to unwanted influences on the vehicle.

5
In this report, guidance schemesfor atmospheric maneuverfor both

deterministic and stochastic cases have been investigated [23, 24, 27].

(iv) Aerocruise

There are basically three methods of plane change, (i) impulsive method,

(ii) aeroglide method, and (iii) aerocruise method. In Impulsive method, the

plane change is achieved entirely outside the atmosphere, and fuel consumption

is prohibitively large for sizable changes of orbital plane. In both aeroglide

and aerocruise methods, rockets are used to deflect the vehicle into the

atmosphere, and the plane change is accomplished by heading change of the

vehicle. With aeroglide there is no thrusting during the atmosphere, and with

aerocruise, atmospheric drag is balanced by a continuous thrust to keep the

spacecraft at a constant altitude and velocity. Propellent expenditure

comparisons among the three methods of plane change show that the aerocrulse

method is superior to other competing methods for plane changes greater than

about 20 degrees, and with heating restraints. The basic effect of propulsion

during aerocruise is to (i) balance drag in order to maintain constant

velocity, (ii) augment lift with a component of thrust, thus increasing

cruising altitude over what it would be during aeroglide turn, and finally

(iii) provide a lateral component of thrust giving the required turn necessary

for plane change.

In this report, research has been conducted into cruise maneuver being

performed using either bank control with constant thrust, or thrust control

with constant bank control [26, 28].

6
II. SUMMARY/ABSTRACT OF RESEARCH WORK

(NASA Langley Grant NAG1-736}
SINGULAR PERTURBATION AND TIME SCALE

e

APPROACHES IN DISCRETE CONTROL SYSTEMS

Dr. D. S. Naidu

Dept. of Elect. and Computer Engineering

Old Dominion University

Norfolk, YA, 23508

and

Dr. D. B. Price

Spacecraft Control Branch

NASA Langley Research Center

Hampton, VA, 23666

Abstract: The theory of singular perturbations and time scales (SPATS) has

been a powerful analytical tool in the analysis and synthesis o£ continuous

and discrete Control systems. In this paper, we first consider- a singularly

perturbed discrete control system. Using singular perturbation approach, outer

and correction subsystems are obtained. Next, by the application of time scare

approach via block diagonallzation transformations, the original system is

decoupled into slow and fast subsystems. To a zeroth order approximation, the

singular perturbation and time scale approaches yield equivalent results.

I{_)ui_hly speaking, the zeroth o['dur a|)proxlmatlon is uomutlmuu callud t,l,u

first approximation. This result is similar to a corresponding result in

continuous control systems.

* See items 1 and 14 under List of Publications

8
SINGULAR PERTURBATIONS AND TIHE SCALES

iN DISCRETE CONTROL SYSTEMS-AN OVERVIEW

Dr. D. S. Naidu

Dept. of Elect. and Computer Engineering

Old Dominion University

Norfolk, VA, 23508

Dr. D. B. Price

Spacecraft Control Branch

NASA Langley Research Center

Hampton, VA, 23665

and

Dr. J. L. Hibey

Dept. of Elect. and Computer Engineering

Old Dominion University

Norfolk, VA, 23508

Abstract: This paper presents an overview of recent developments in the theory

of singular perturbations and time scales (SPATS) in discrete control systems.

The focus is in three directions: modeling, analysis, and control. First,

sources of discrete models and the effect of the dlscretizing interval on the

modeling are reviewed. Then the analysis of two-time scale systems is

pru_cnted to bI'Ing out typical chavacteulstlc features oi" SPATS. Finally, in

the control of the two-time scale systems, we the important issue of

multirate sampling is addressed. The bibliography containing over 100 titles

is included.

" See items 2 and 8 under List of Publications

9
O

ON THE METtlOD OF MATCI[ED ASYMPTOTIC EXPANSIONS

Dr. D. S. Naidu

Dept. of Elect. and Computer Engineering

Old Dominion University

Norfolk, VA, 23S08

and

Dr. D. B. Price

Spacecraft Control Branch

NASA Langley Research Center

Hampton, VA, 23665-5226

Abstract: A critical examlnatlon of the method of matched asymptotic

expansions (MAE) reveals that the various terms of tile common solution oi" HAg

can be generated as polynomials In stretched varlable without actually solving

for them from the outer solution as is done presently. This also shows that

the common solution of the method of MAE and the Intermediate solution of

singular perturbation method are the same and hence that these methods give

identical results for a certain class of problems. An illustrative example is

given.

* See items 3, 7 and 18 under List of Publications

10
IMPACT OF ATMOSPHERIC DENSITY SCALE HEIGHT ON THE PERFORMANCE

e

OF AEROASSISTED COPLANAR ORBITAL TRANSFER VEHICLES

Dr. D. S. Naldu

Dept. of Elect. and Computer Engineering

Old Dominion University

Norfolk, VA, 23508

and

Dr. D. B. Price

Spacecraft Controls Branch

NASA Langley Research Center

Hampton, Virginia, 23665

Abstract: A common way of representing atmospheric density is by a,,

exponential form using scale height, which is assumed to be constant over the

whole interval of atmospheric altitude. In this simulation, the scale height

h_s been readjusted depending upon the altitude Intel'val, and slmul_tlons arc

c_i'ried out for an aeroasslsted, coplanar orbital transfer vehicle.

" See item 4 under List of Publications

II
TIME-SCALE SYNTHESIS OF A CLOSED-LOOP DISCRETE OPTIMAL CONTROL SYSTEM

Dr. D. S. Naldu

Dept. of Elect. and Computer Engineering

Old Dominion University

Norfolk, VA 23529

and

Dr. D. B. Price

Spacecraft Controls Branch

NASA Langley Research Center

Hampton, VA 23665-5225

Abstract: A two-time-scale discrete control system is considered. The

closed-loop optimal linear quadratic regulator for the system requir'es the

solution of a full-order algebraic matrix Riccati equation. Alternatively, the

original system is decomposed into reduced-order slow and fast subsystems. The

closed-loop optimal control of the subsystems requires the solution of two

algebraic Riccati equations of an order lower than that required for the

full-order system. A composite, closed-loop suboptimal control is created from

the sum of the slow and fast feedback optimal controls. Numerical results

obtained for an aircraft model show a very close agreement between the

exact(optimal) solutions and computatlonally simpler composite(suboptimal)

solutions. The main advantage of the method is the considerable reduction in

the overall computational requirements for the closed-loop optimal control of

digital flight systems.

See item 6 under List of Publications

12
THREE-DIMENSIONAL ATMOSPHERIC ENTRY PROBLEM

USING METHOD OF MATCHED ASYMPTOTIC EXPANSIONS

Dr. D. S. Naidu

Dept. of Elect. and Computer Engineering

Old Dominion University

Norfolk, VA, 23508

Abstract: Tile analysis of a three-dimensional atmospheric entry problem using

the method of matched asymptotic expansions Is considered. A composite

solution is formed in terms of an outer solution, an inner solution and a

common solution. The outer solution Is obtained from gravitationally dominant

region, whereas the aerodynamically dominant region contributes to the inner

solution. The common solution accounts for the overlap between the outer and

inner regions. In comparison to the previous works, the present simplified

methodology yields explicit analytical expressions for various components of

the composite solution without resorting to any type of transcendental

equations to be solved only by numerical methods. The method is applicable for

obtaining autonomous guidance and control strategies for a variety of

aerospace vehicles.

J See items 9, 11 and 22 under List of Publications

13
FUEL-OPTII_L TRAJECTORIES FOR AEROASSISTED

g

COPLANAR ORBITAL TRANSFER PROBLEM

Dr. D. S. Naldu, Dr. J. L. Hlbey, and C. Charalambous

Dept. of Electrical and Computer Engineering

Old Dominion University

Norfolk, VA, 23529

Abstract: We first describe briefly the various types of coplanar transfers.

Then we address the fuel-optimal control problem arising in coplanar orbital

transfer employing aeroasslst technology. The maneuver Involves a transfer

from high Earth orbit to low Earth orbit and at the same time minimization of

the fuel consumption for achieving the deslred orbit transfer. It is known

that a change in velocity, also called the characteristic velocity, is a

convenient parameter to measure the fuel consumption. A suitable performance

index is the total characteristic velocity which is the sum of the

characteristic velocities for deorblt and for reorblt (or clrcularization),

Use of Pontryagln mlnlmum principle leads to a nonlinear, two-point boundary

value problem in state and costate variables. The solution of the TPBVP is the

stumbling block in obtaining fuel-optimal solution. This problem Is solved by

using a more efficient multiple shooting method which is a simultaneous

application of a single shooting algorithm to equally divided points of the

total interval of the solution.

" See items 12, 17 and 25 under List of Publications

14
FUEL-OPTIM_TRAJECTORIES OF AEROASSISTED

o

ORBITAL TRANSFER WITH PLANE CHANGE

Dr. D. S. Naldu

Dept. of Elect. and Computer Engineering

Old Dominion University

Norfolk, VA, 23529

Abstract: The fuel-optimal control problem arising In noncoplanar orbital

transfer employing aeroassist technology is addressed. The mission involves

the transfer from high Earth orbit to low Earth orbit with plane change. The

complete maneuver consists of a deorblt impulse to inject a vehicle from a

circular orbit to elliptic orbit for the atmospheric entry, a boost impulse at

the exit from the atmosphere for the vehicle to attain a desired orbital

altitude and finally a reorblt impulse to circularize the path of the vehicle.

In order to minimize the total fuel consumption, a performance index is chosen

as the sum of the deorbit, boost, and reorblt impulses. Application of

optimization principles leads us to a nonlinear, two-point, boundary value

problem, which is solved by using a multiple shooting method.

" See items 13, 19, 20 and 29 under List of Publications

IS
FUEL-OPTIMAL TRAJECTORIES FOR NONCOPLANAR

ORBITAL TRANSFER VEHICLES

Dr. D. S. Naldu

Dept. of Elect. and Computer Engineering

Old Dominion University

Norfolk, VA 23529

ABSTRACT: The fuel-optimal problem In noncoplanar orbital transfer employing

aeroassist technology is addressed. The mission involves the transfer from

high Earth orbit to low Earth orbit with plane change. The complete maneuver

consists of a deorbit impulse to inject a vehicle from a circular orbit to

elliptic orbit to enter the atmosphere, a boost impulse at the exit from the

atmosphere for the vehicle to attain a desired orbital altitude and finally a

reorbit impulse to circularize the path of the vehicle. In order to minimize

the total fuel consumption, a performance index Is chosen as th_: sum of" the

d_orblt, boost, and reorblt Impulses. For _ typlc_l _uro_slsLud ol'blLul

transfer vehicle with high llft-to-drag ratio, the simulations are carried out

using industry standard program to optimize simulated trajectories (POST).

" See item 15 ur_der Lis_ oF Publications

16
SINGULAR PERTURBATIONS AND TIME SCALES

I

IN THE DESIGN OF DIGITAL FLIGHT CONTROL SYSTEMS

Dr. D. S. Naldu

Dept. of Elect. and Computer Engineering

Old Dominion University

Norfolk, VA 23529

and

Dr. Douglas B. Price

Spacecraft Controls Branch

NASA Langley Research Center

Hampton, VA 23665-5226

Abstract: This paper investigates the application of methodology of singular

perturbations and time scales (SPATS) to the control of digital flight

systems. A block diagonalization method is developed to decouple a full-order,

two-time (slow and fast) scale, discrete control system into reduced-order

slow and fast subsystems. Basic properties and numerical aspects of the method

are explored. This reveals an interesting fact that singularly perturbed

discrete systems can be viewed as two-time scale systems. Next, the

closed-loop optimal control of the two-time scale full-order system involves

the solution of a full order algebraic matrix RiccatI equation. Alternatively,

using the block diagonalization method, the full-order system is decomposed

into reduced-order slow and fast subsystems. The closed-loop optimal control

of the subsystems requires the solution of only reduced-order algebraic matrix

}_Iccatl equations. A composite closed-loop suboptimal control is constructed

as the sum of the slow and fast optimal feedback controls. Numerical

experimentation with an aircraft model shows close agreement between the exact

solutions and the decoupled (or composite) solutions. The main advantage of

the method is the considerable reduction in the overall onboard computational

requirements for the evaluation of optimal guidance and control laws. It is

believed that this paper also serves as a source of brief survey of digital

flight systems.

See item 16 under List of Publications

17
NEIGHBORING OPTIMAL GUIDANCE FOR AN AEROASSISTED ORBITAL

TRANSFER VEHICLE IN THE PRESENCE OF MODELLING UNCERTAINTIES

Dr. D. S. Naldu, Dr. J. L. Hlbey, and C. Charalambous

Dept. of Electrical and Computer Engineering

Old Domlnlon University

Norfolk, VA, 23529

ABSTRACT: We Intend to devise a neighboring optimal guidance scheme fur a

nonlinear dynamic system with stochastic inputs and perfect measurements as

applicable to fuel optimal control of an aeroasslsted orbital transfer

vehicle. For the deterministic nonlinear dynamic system describing the

atmospheric maneuver, a nominal trajectory is determined. Then, a nelghboring,

optimal guidance scheme is obtained. Taklng modelling uncertalntles into

account, a linear, stochastic, neighboring optimal guidance scheme is devised.

Finally, the optimal trajectory is approximated as the sum of the

deterministic nominal trajectory and the stochastic nelghborlng optimal

solution. Numerical results are presented for a typical vehicle.

" See items 23 and 27 under List of Publications

18
NEIGHBORING OPTIMAL GUIDANCE

m

FOR AEROASSISTED NONCOPLANAR ORBITAL TRANSFER

D. S. Naldu

Dept. of Elect. and Computer Engineering

Old Dominion University

Norfolk, VA, 23508

Abstract: The fuel-optimal control problem in aeroassisted noncoplanar orbital

transfer Is addressed. The equations of motion for the atmospheric maneuver

are nonlinear and the optimal (nominal) trajectory and control are obtained.

In order to follow the nominal trajectory under actual conditions, a

neighboring optimum guidance scheme is designed using linear quadratic

regulator (I.QR) theory for onboard real-tlme Implementation. One of the stat_

variables is used as the independent variable in preference to the time. The

weighting matrices In the performance index are chosen by a combination of a

heurlstlc method and an optimal modal approach. The necessary feedback conti'ol

law is obtained in order to minimize the deviations from the nominal

conditions.

" See item 24 under List of Publlcatlons

19
O

ORBITAL PLANE CHANGE MANEUVER WITH AEROCRUISE

Dr. D. S. Naldu

Dept. of E1ectrlcal and Computer Engineering

Old Dominion University

Norfolk, VA

Abstract: The synergistic plane change problem connected with orbital trunsrul"

employing aeroasslst technology, is addressed. The mission involves transf'er

from high Earth orbit to low Earth orbit wlth plane change being performed

within the atmosphere. The complete mission consists of a deorbit phase,

atmospheric phase, and finally reorblt phase. The atmospheric maneuver is

composed of an entry mode, a cruise mode, and finally an exit mode. During the

cruise mode, constant altitude and velocity are maintained by means of bank

angle control with constant thrust or thrust control wlth constant bank angle.

Comparisons between these two control strategies bring out some interesting

features.

* See items 26 and 28 under List of Publications

20
Ill. LIST OF PUBLICATIONS

(NASA Langley Grant NAG1-736}

1. D. S. Naidu and D. B. Price, "Singular perturbation and time scale

approaches in discrete control systems", in Research Report, Dept. of Elect.

and Computer Engineering, Old Dominion University, Norfolk, VA, June 1987.

2. D. S. Naidu, et. _1., "Singular perturb_tlons and time scales i, discr'ete

control systems-an overview", in Research Report, Dept. of Elect. and Computer"

Engineering, Old Dominion University, Norfolk, VA, June 1987.

3. D. S. Naidu, and D. B. Price, "On the method of matched asymptotic

_xpauslot_", in Reseat'oh }_eport, Dept. ot" Eject. and Computer" EngineeFin_, Old

Dominion University, Norfolk, VA, June 1987.

4. D. S. Naidu and D. B. Price, "Impact of atmospheric scale height on the

performance of aeroassisted orbiter transfer vehicles", in Research Report,

Dept. of Elect. and Computer Engineering, Old Dominion University, Norfolk,

VA, June 1987.

5. L. W. Taylor, Jr., and D. S. Naidu, "Experience in distributed parameter

modeling of the spacecraft control laboratory experiment (SCOI.E) structure",

AIAA Dynamics Specialists Conference, Monterey, CA, April 1987.

6. D. S. Naidu and D. B. Price, "Time scale synthesis of a closed-loop

discrete optimal control system", Journal of Guidance, Control, and Dynamics,

10, 417-421, Sept.-Oct. 1987.

7. D. S. Naidu, and D. B. Price, "On the method of matched asymptotic

expansions", SIAM Annual Meeting and 35th Anniversary, Denver, CO, October

12-15, 1987

8. D. S. Naldu, el. al., "Singular perturbations and time scales in discrete

control systems-an overview", Survey Paper at the Invited Session, IEEE

Conference on Decision and Control, Los Angles, CA, Dec. 9-11, 1987

21
9. D. S. Naidu, "There-dlmenslonal atmospheric entry problem using method of

matched asymptotic expansions", in Research Report, Dept. of Elect. and

Computer Engineering, Old Dominion University, Norfolk, VA, December 1987.

10. l). S. Naidu, Singular Perturbation Methodology In Control Systems, ILL

Control Engineering Series, Peter Peregrlnus Ltd., Stevenage Hefts, England,

1988.

11. D. S. Naldu, "There-dimensional atmospheric entry problem using method of

matched asymptotic expansions", 1988 American Control Conference, Atlanta, GA,

June 14-17, 1988.

12. D. S. Naldu, et. al., "Fuel-optimal trajectories for aeroassisted,

coplanar, orbital transfer problem", Research Report, Dept. of Elect. and

Computer Engineering, Old Dominion University, Norfolk, VA, June, 1988.

13. D. S. Naldu, "Fuel-optimal trajectories for aeroassisted, noncoplanar,

orbital transfer problem", Research Report, Dept. of Elect. and Computer

Engineering, Old Dominion University, Norfolk, VA, June, 1988.

14. D. S. Naidu and D. B. Price, "Singular perturbation and time scale

approaches in discrete control systems", Journal of Guidance, Control and

Dynamics, Vol., 11, no. 5, pp. 592-594, Nov.-Dec. 1988.

15. D. S. Naidu, "Fuel-optimal trajectories for noncoplanar orbital transfer

vehicles", Research Report, Dept. o£ Elect. and Computer Engineering, Old

Dominion University, Norfolk, VA, December 1988.

16. D. S. Naidu and D. B. Price, "Singular Perturbations and Time Scales in

the Design of Digital Flight Control Systems", NASA Technical Paper 2844,

Langley Research Center, Hampton, December 1988

17. D. S. Naldu, et. al., "Optimal control of aeroasslsted, coplanar, orbital

transfer vehicles", 26th IEEE Conference on Decision and Control, Austin,

Texas, December 7-9, 1988.

22
18. D. S. Naldu and D. B. Prlce, "On the method of matched asymptotic

expansions", Journal of Guidance, Control, and Dynamics, 12, 277-279,

March-April 1989.

19. D. S. Naidu, "Fuel-optimal trajectories of aeroassisted orbital transfer

with plane change," In Research Report, Dept. of Elect. and Computer

Engineering, Old Dominion University, Norfolk, VA, June 1989.

20. D. S. Natdu, "Fuel-optimal trajectories of aeroassisted orbital transfer

with plane change," AIAA Guidance, Navigation, and Control Conference,

Boston, MA, August 14-16, 1989.

21. D. S. Naldu, and Anthony J. Callse, "Singular perturbations and time

scales in control theory and applications: survey 1983-1989", IFAC Workshop on

Singular Perturbations and Asymptotic Methods In Systems and Control, Boston,

HA, Aug. 17-18, 1989.

22. D. S. Naidu, "There-dimenslonal atmospheric entry problem using method of

matched asymptotic expansions", IEEE Trans. on Aerospace and Electronic

Systems, 25, 660-667, Sept. 1989.

23. D. S. Naidu, et. at., "Fuel-optlmal trajectories for aeroassisted coplanar

orbital transfer vehicles In the presence of uncertainties due to modelling

inaccuracies", in Research Report, Dept. of Electrical and Computer

Engineering, Old Dominion University, Norfolk, VA, December 1989

24. D. S. Naidu, "Neighboring optimal guidance for aeroasslsted noncoplanar

orbital transfer", In Research Report, Dept. of Electrical and Computer

Engineering, Old Dominion University, Norfolk, VA, December 1989

25. D. S. Naidu, et. al., "Fuel-optlmal trajectories for aeroassisted coplanar

orbital transfer problem", IEEE Trans. on Aerospace and Electronic Systems,

26, 374-381, March 1990.

26. D. S. Naidu, "Orbital plane change maneuver with aerocruise", in Research

Report, Dept. of Elect. and Computer Engineering, Old Dominion University,

Norfolk, VA July, 1990

23
27. D. S. Naldu, et. al., "Nelghborlng optimal guldance for an aeroassisted

orbital transfer vehicle in the presence of modelling uncertainties", AIAA

Guidance, Control, and Navlgatlon Conference, Portland, OR, August 20-22, 1990

28. D. S. Naldu, "Orbital plane change maneuverwith aerocrulse", accepted for"

presentation at AIAA 29th AerospaceScience Meeting and Exhibit, Reno, Nevada,

January 7-10, 1991.

29. D. S. Naidu, "Fuel-optimal trajectories of aeroassisted orbital transfer

with plane change," IEEETrans. Aerospaceand Electronic Systems, 27, May 1991

(in press).

24

GUIDANCE AND CONTROL STRATEGIES FOR AEROSPACE VEHICLES

(NASA Langley Grant NAGI-736)

PART II

This final report under NASA Langley grant NAGI-736 consists of two

parts. Part I consists of the summary of the earlier work whose reports have

been submitted at various times during the period (I/I/87 to 12/31/1989). Part

[I describes in detail the research work done during the period I/I/1990 to

?/31/1990.
ORBITAL PLANE CHANGE MANEUVER WITH AEROCRUISE

Dr. O. S. Naidu

Dept. of Electrical and Computer Engineering

Old Dominion University

Norfolk, VA

AtJ<l_'os_ For coH'u_pond_nc_: I)r. D. S. Naldu

Spacecraft Control Branch, H_ 161

NASA Langley Research Center

Hampton, VA, 23665-5225

Ph: (804) 864-6498
TABLE OF CONTENTS

Io

II.

Ill.

IV.

V.

VI.

VII.

VIII.

IX.

X.

Pa_e

AFIS'FRACT ............................................ 2

NOMENCLATURE ........................................ 2

I NTRODUCT I ON ........................................ 4

MISSION DESCRIPTION ................................. 7

DESCENT MODE........................................ 9

AEROCRUISE MODE: BANK ANGLE CONTROL ................ 14

AEROCRUISE MODE: THRUST CONTROL .................... 22

ASCENT HODE ........................................ 26

BOOST AND REORBIT PHASE ............................ 29

NUMERICAL DATA AND RESULTS ......................... ,30

CONCLUDING REMARKS ................................. :34

REFERENCES ......................................... :35

i.i
LIST OF FIGURES

1. Aeroassisted Orbltal Transfer Mission ................................ 37

2. Atmospheric Phase .................................................... 38

3. Coordinate System .................................................... 39

4. Lift and Drag Coefficients ........................................... 40

5. Lift-to-Dra R Ratio and Aeropropulslve Efficiency ..................... 41

6(a). Altitude as a Function of Time during Descent Mode ................ 42

5(b). Velocity as a FuncLlon of Time during Descent Mode ................ 43

6(c). Flight Path Angle as a Function of Time During Descent Mode ....... 44

?. Optimum Thrust Angle as a Function of Angle of Attack ................ 45

8. Maximum Heading Angle as a Function of Angle of Attack ............... 46

9(a). Mass as a Function of Time with Bank Angle Control ................ 4?

9(b). Bank Angle as a Function of Time with Bank Angle Control .......... 48

9(c). Heading Angle as a Function of Time with Bank Angle Control ....... 49

10. Optimum Bank Angle as a Function of Angle of Attack ................. 50

11. Maximum Heading Angle as a Function of Angle of Attack .......... 51

12(a). Mass as a Function of Time with Thrust Control ................... 52

12(b). Thrust as a Function of Time with Thrust Control ................. 53

12(c). Thrust AllEle as a Function o£ Time with Thl'tl_t Contl'ol ........... 54

12(d). Heading Angle as a Function of Time with Thrust Control .......... 55

13. Maximum Heading Angle as a Function of Angle of Attack .............. 5G

14(a). Altitude as a Function of Time During Ascent Mode ................ 5?

14(b). Velocity as a Function o£ Time During Ascent Mode ................ 58

14(c). Flight Path Angle as a Function of Time During Ascent Mode ....... 59

III
1S(a}. Altitude as a Function of Time with Hank Angle Control ........... 60

15(b). Velocity as a Function of Time with Bank Angle Control ........... 61

15(c). Flight Path Angle as a Function of Time with Bank Angle Control..62

15(d}. Heading Angle as a Function of Time with Bank Angle Control ...... 63

15(e). Heat Rate as a Function of Time with Bank Angle Control .......... 64

16(a). Altitude as a Function of Time with Thrust Control ............... 65

16(b). Velocity as a Function of Time with Thrust Control ............... 66

16(c). Flight Path Angle as a Function of Time with Thrust Control ...... 67

16{d). Heading Angle as a Function of Time with Thrust Control .......... 68

16(e). tleat Rate as a Function of Time with Thrust Control .............. 6S

iv
ORBITAL PLANE CHANGE MANEUVER WITH AEROCRUISE

Dr. D. S. Naidu

Dept. of Electrical and Computer Engineering

Old Dominion University

NorFolk, VA

Address for correspondence: Dr. D. S. Naidu

Spacecraft Control Branch, HS 161

NASA Langley Research Center

Hampton, VA. 23665-5225

Ph: (804) 864-8498
Abstract: The synergistic plane change problem connected with orbital transfer

employing aeroassist technology, is addressed. The mission involves transfer

from high Earth orbit to low Earth orbit with plane change being performed

within the atmosphere. The complete mission consists of a deorbit phase,

atmospheric phase, and finally reorbit phase. The atmospheric maneuver is

composed of an entry mode, a cruise mode, and finally an exit mode. During the

cruise mode, constant altitude and velocity are maintained by means of bank

angle control with constant thrust or thrust control with constant bank angle.

Comparisons between these two control strategies bring out some interesting

features.

Nomenclature

C :

D

C :

DO

C :

L

C :

LR

D :

E :

E :

P

g :

H :

:

I :

sp

i :

J :

K :

L :

m :

Q :

R :

R :

g

S :

T :

t :

drag coefficient

drag coefficient at zero lift

lift coefficient

lift coefficient for maximum llft-to-drag ratio

drag force

maximum value of lift-to-drag ratio

aeropropulsive efficiency

gravitational acceleratlon

altitude

Hamiltonian

specific fuel consumption

inclination

performance index

induced drag factor

lift force

vehicle mass

Heating rate

distance from Earth center to vehicle center of gravity

radius of Earth

aerodynamic reference area

thrust

time
V : velocity

angle of attack

inverse atmospheric scale height

flight path angle

normalized density

= normalized llft coefficient

: thrust angle

8 : down rangeangle

l : costate (Langrange) variable

: gravitational constant of Earth

p : density

: bank angle

¢ : cross range angle

¢ : heading angle

AV : characteristic velocity

Subscripts

a : atmospheric boundary

c : clrcularizatlon at LEO

d : deorbit at HEO

e : entry to atmosphere

f : exit from atmosphere

j : beginning of aerocruise

n : end of aerocruise

s : surface level
I. INTRODUCTION

The main function of space transportation system is to deliver payloads

from Earth to various locations in space. Until now, this function has been

performed by various rockets, the space shuttle, and expendable upper stages

using solid or liquid propellants. In particular, considering the economic

benefits and reusability, an orbital transfer vehicle (OTV) is proposed for

transporting payloads between low Earth orbit (LEO) and high Earth orbit

(HEO).I The two basic operating modescontemplated for OTVape a ground-based

OTVwhich returns to Earth after each mission and a space-based OTVwhich

operates out of an orbiting hanger located at the proposed SpaceStation

Freedom.

In a typical mission, a space-based OTV, which Is initially at the space

station orbit (SSO), is required to transfer a payload to geosynchronousEarth

orbit (GEO), pick up another payload, say a faulty satellite, and return to

rendezvous with the orbiting hanger at SSOfor refurbishment and redeployment

of payload. The OTVon its return journey from GEOto SSOneeds to dissipate

someof its orbital energy. This can be accomplished by using an entirely

propulsive (Hohmann)transfer in space only or a combination of propulsive

transfer in space and aeroassisted maneuverin the atmosphere. It has been

established that a significant fuel savings and hence increased payload

capabilities can be achieved with propulsive and aeroasslsted maneuvers

I

instead of all-propulsive maneuvers. Thls leads to an aeroasslsted orbital

transfer vehicle (AOTV), which on its return leg of the mission, dips into the

Earth's atmosphere, utilizes atmospheric drag to reduce orbital velocity and

to achieve a desired orbital inclination. Basically, the AOTV performs a

synergistic maneuver, employing a hybrid combination of propulsive maneuver in

space and aerodynamic maneuver in the atmosphere.

The plane change capability is required to (i) orbit a vehicle in a plane

which does not pass through a launch site, (li) shorten the time needed to

reach multiple reconnaissance targets on a single orbital mission, (lii)

reduce the time needed to return to base from orbit, (iv) perform effective

rendezvous with satellites in different orbital planes, (v) avoid flights over

hostile territory, and finally (vi) facilitate arrival and departure flights

4
from Space Station Freedom, in fulfilling specified mission objectives. 2 It

should be noted that an orbital plane is usually defined in terms of

inclination and longitude of the ascending node. For our present purpose only

an inclination change is controlled.

There are basically three methods of plane change, (i) impulsive method,

(ii) aeroglide method, and (iii) aerocrulse method. In impulsive method, the

plane change is achieved entirely outside the atmosphere, and fuel consumption

is prohibitively large for sizable changes of orbital plane. In both aeroglide

and aerocruise methods, rockets are used to deflect the vehicle into the

atmosphere, and the plane change is accomplished by heading change of the

vehicle. With aeroglide there is no thrusting during the atmosphere, and with

aerocruise, atmospheric drag is balanced by a continuous thrust to keep the

spacecraft at a constant altitude and velocity. Propellent expenditure

comparisons among the three methods of plane change show that the aerocrulse

method is superior to other competing methods for plane changes greater than

about 20 degrees, and wlth heating restraints. The basic effect of propulsion

during aerocruise is to (i) balance drag in order to maintain constant

velocity, (ii) augment lift with a component of thrust, thus increasing

cruising altitude over what it would be during aeroglide turn, and finally

(iii] provide a lateral component of thrust giving the required turn necessary

for plane change. The aeroglide and aerocruise methods utilizing atmospheric

maneuver in conjunction with propulsion augmentation are also termed the

"synergistic" or "aeropropulslve '° methods. 3'4

5-7

The following are some of the features of atmospheric plane change.

(i) For plane changes of less than 15 degrees, an all-propulsive maneuver is

generally more efficient. (ii) An L/D of at least 2 is required to offer a

significant advantage over the all-propulsive plane change, and it is

desirable to maximize the L/D of a vehicle. (ill) A plane change made at a

node produces all inclination change whereas a turn at an orbit apex (90

degrees from node) provides no inclination change, only a shift in the node.

Hence, for maximum inclination change and minimum node shift, the turn should

be centered over the node in the shortest possible duration. Thus, plane

changes performed at maximum C (i.e., high angle of attack) which are quicker

L

are more fuel efficient than the slower maximum L/D turns. (iv) The total heat
load can be reduced substantially by carrying out a quicker high

angle-of-attack turns rather than the slower maximumL/D turns, iv) An

aerocruise (thrusting) turn offers significant advantages over an aeroglide

(non-thrusting) turn, whenthe desired plane change is more than i0 degrees.

(vi) During aerocruise, the high angle-of-attack and bank attitude of the

vehicle produce a lateral componentof thrust, which is responsible foP a

significant amount of plane change.

Comparedto other works,s-8 the highlights of the present work are (i)

The analysis covers the complete mission from HEOto LEO. {il) The descent

mode and ascent mode of atmospheric phase are analyzed with flight path angle

as an independent variable. (iii) The time has been retained as an independent

variable duping the cruise mode. (iv) DuPing the cruise mode, both bank angle

control and thrust control are analyzed.

In this report, we address the synergistic plane change problem arising

in noncoplanar orbital transfer employing aeroasslst technology. The mission

involves the transfer from HEO to LEO with plane change being performed within

the atmosphere. The complete mission consists of a deorblt phase, an

atmospheric phase, and finally a reorblt phase. The atmospheric maneuver phase

is composed of descent (entry) mode, cruise mode, and ascent (exit) mode.

During the aerocruise mode, constant altitude and velocity are maintained

either by (i) varying bank angle with constant thrust, or by (ll) varying

thrust with constant bank angle. The comparison of these two control schemes

bring out some interesting features. Numerical results are given for typical

data.

6
II. MISSIONDESCRIPTION

For an orbital transfer problem, the following assumptions are made. (i)

The initial HEOand final LEOorbits are circular. (il) The mission is

comprised of three impulses. (ill) The vehicle is represented as a constant

point massduring atmospheric pass. (iv) A Newtonian inverse square

gravitational field is used. (v} Earth's rotation is neglected. (vi) The

atmosphere is exponential.

The complete mission from HEOto LEOwith atmospheric pass is depicted in

Fig. I. It consists of a deorbit phase, an atmospheric phase, and a reorbit

phase. There are three impulses: first, a deorbit impulse AV at HEOto inject d

a vehicle into a HEO-entry elliptic orbit, second, a boost impulse AV at the b

exit from the atmosphere for the vehicle to attain sufficient velocity to

travel along an exit-LEO elliptic orbit, and finally, a circularizing impulse

AV to circularize the path of the vehicle. The atmospheric phase itself is

C

composed of descent (entry) mode, cruise mode, and ascent (exit) mode.

Consider the basic equations of motion for different phases of deorbit,

aeroassist (or atmospheric flight), boost and reorbit (or circularlzation).

Deorbit Phase

Initially, we assume that a spacecraft is in a circular orbit of radius

Rd, well outside the Earth's atmosphere, moving with a circular velocity V d =

_V_ d. Deorbit is performed by means of an impulse AVd, to transfer the

vehicle from the circular orbit to elliptic orbit with perigee low enough to

intersect the dense part of the atmosphere [Fig. 1]. At D, since the elliptic

velocity is less than the circular velocity, the impulse AV is executed so as

d

to oppose the circular velocity V . The deorbit impulse AV causes the vehicle

d d

to enter the atmosphere of radius R with a velocity V and flight path angle

a

y . It is known that the optimal-energy loss maneuver from the circular orbit

e

is simply the Hohmann transfer and the impulse is parallel and opposite to the

instantaneous velocity vector.

Using the principle of conservation of energy and angular momentum at the
deorblt point D, and the atmospheric entry point E, we get,

v2/2 - _./R : (V - AVd)_'2- ./R d (1)

R V cos(-7 ) = R (V d - AV ) a e e d d

(2)

from which solving for AVd, we get

hVd : 4.,Rd- J2._i/Ra - 1/_)/[(RJR )2/cos'_e - i] (3)

It is easily seen that the minimum value of the deorbit impulse AV

dm

obtained at _" = O, corresponds to an ideal transfer wherein the space vehicle

e

grazes along the atmospheric boundary. To ensure proper atmospheric entry, the

deorbit impulse AV d must be higher than the minimum deorbit impulse AVdm which

is given by

_Lm:_r_d-]2_ci_R-I_R)/[CRJR)2-i] C4)

Aeroassist (Atmospheric) Phase

The atmospheric phase of the mission is composed of (i) descent mode,

(li) cruise mode, and (ill) ascent mode [Fig. 2].
III DESCENT MODE

During the descent mode, the equations of motion for the vehicle (without

Io

any thrusting) are given below [Fig. 3]. The kinematic equations are,

dR

- Vsin_ (5a)

dt

de _ Vcos_cos_/Rcos¢ (5b)

dt

d¢ _ Vcos_sin_/R (5c)

dt

The force equations are

dV

mdt - D - mgsin_ (5d)

d_

mV_-{ = Lcosc + m(V_R - g)cos_

mV_-{ = Lsinc/cos_ - (mV_R)cos_cos@tan¢

where,

L = CL(_)pSV2/2" , D = CD( _) pSV2/2; CD = CDO + KC2L

g = _/R2; • exp(-_H) R = H + RE, p = Ps

(Se)

Csf)

Neglecting mass terms in comparison to aerodynamic terms in (5), we get

dH

- Vsin_ dt

(6a)

dV

- D/m

dt

(6b)

d_ _ Lcosv/mV

dt

(6c)

9
d_

dt - Lstn_/mVcos_ (6d)

Using flight path angle as the independent variable, we get

dH

d_ - 2mexp(_H)sin_/PoSCLcosv (7a)

dV

d7 CDV/CLC°S_ (7b)

d@ _ tanv/cos_ (7c)

d_

During the descent mode, let us assume that there is no banking of the

vehicle, and hence no heading is achieved. Then (7) becomes

dH

d--{= 2mexp(_H)sin_/PoSC L (8a)

dV

_= - C V/C (8b)

dT D U

Optimal Control Problem

The optimal control problem is posed as follows. Given entry conditions,

and the conditions at the end of the descent mode (or the initiation of

aerocruise mode), find the optimal control law which maximizes the final

11

velocity, subject to altitude constraint H z H . This altitude constraint

b

implies in a way heat-rate constraint. The performance index is given by

J = - V (9)

b

The Hamiltonian for (8) and (9) is

(IO)

The adjoint equations are

10
dAH/d_ = -A H2m_exp(_H)sin_/PoSC L

(lla)

dA /d_ = A C /C

v vD L

(llb)

Solving the state (8) and costate (II) equations, we get

VA = V A = constant

V e ve

(12a)

A exp(r3H) = A exp(_H )

H He e

The boundary conditions for the adJoint variables are

aJ 1 = - 1

(12b)

(13a)

j) aJ I = o

(13b)

From (12) and (13),

;_ = 0

H

With (14), the Hamiltonian (I0) reduces to

R = - _ V[C /C + KC ]

v DO L L

(14)

(15)

The optimal control is then given by

aR _ o (16)

ao

L

leading to

c = 4-d /K= c (17)

LO DO LE

where, CUE is the lift coefficient for maximum lift-to-drag ratio (L/D)_x = E

11
= 1/24KC

DO

altitude as

With optimal control (17) in (8), we solve for the velocity and

V(_) = V exp[-(_-_ )/E] (18)

e e

_xp(_flHe 2Bm H = In ) +

, PoSCLo

_(COS_ - COS_e)] -1/_

For small 7, (19) reduces to

H = in xp(-13H ) grn (7 - _" )

e PoSCLo

(19a)

(19b)

At the start of the descent mode, _ = _ , and at the end of the descent mode

e

(or the beginning of the aerocruise mode), _ = _j = O. Then the above

relations become

]-1/_ Hj = in[exp(-/3H e) + PoS_2Bm (I -cos_' e) (19c)

LO

and with the approximation,

+ _m 2]-1/# Hj = In exp(-BH e)

(19d)

Then, the inequality constraint on altitude H z Hi, with (19a) transforms to

and with approximate solution (19b),

_'e poSC or , ] - -_ Lexpt-BHJ ) - exp(-{'JHe)

(2Oh)

12
The velocity at the end of the descent mode is obtained from (18) with _" = _]

= 0 as

= V exp [2 _'e_ (21) Vj e

13
IV. AEROCRUISE MODE: BANK ANGLE CONTROL

We first write down the general equations of motion, inject the

conditions for cruise flight, use the assumptions of small latitude, and

finally optimize the heading change. During aerocruise mode, there is

continuous thrusting. Thus the kinematic equations are [Fig. 3] 10

dH _ Vsin_ (22a)

dt

de _ Vcos_cos¢/Rcos¢ (22b)

dt

de _ Vcos_sin@/R (22c)

dt

The force equations are

dV (22d)

m_-{ = Tcosn - D - mgsin_

mV-_-{ = (Tsinn + L)cos_ + m(V_R - g)cos_

(22e)

d_ (Tsinn + L)sin_/cos_ - (mV_R)cos_cosCtan¢

mVE_=

The propulsion (thrusting) equation is

(22f)

dm _ T/gl (22g)

dt sp

From the above equations of motion, we see clearly that during the

atmospheric maneuver, if the lift vector L is rotated about the velocity

vector V through the bank angle _, it creates a lateral force component {Tsin_

+ L)sinv orthogonal to the vertical plane that has the effect of changing the

heading angle ¢. At the end of the atmospheric phase, the equations (22c) and

1o

(22f) for the cross range angle ¢, and the heading angle _, become,

14
d¢/dt _ tan@

d¢/dt tan¢

(23a)

integration of which yields,

cos¢cos¢ = cosi (23b)

where, i is the Orbital inclination. For small values of cross range angle ¢,

the orbital inclination i is given by the heading angle ¢ itself. Thus, the

total change in the heading corresponds to the change in orbital inclination

(plane change).

Now let us insert the cruise conditions of constant a]t]tude and

velocity. 8 The constant altitude condition on (22a) gives zero flight path

angle throughout. The constant velocity condition on (22d) boils down to

Tcosn= D = pSV2C (_)/2 (24a)

D

We note that the conditions at the beginning of the aerocruise mode are

denoted by the subscript j. However, for simplicity in notation we shall

continue to use the variables without any subscript to denote the cruise

conditions. If the angle of attack e is held constant, then the drag force D

is constant at a constant cruising altitude. Also, since the flight path angle

is zero throughout, (22e) reduces to

(Tsinn + L)cos_ = m(E - VZ/R)

Combining (24a) and (24b), we get

m(g - V2/a)cosn

cosc= D[sinn + (L/D)cosn]

where,

sin_ + (t/D)cos_ = {L + Tsinn)/T = E

p

(24b)

(2S)

(26)

lS
is called aeropropulsive efficiency. Alternatively,

(Dtann + L)cos_ = m[g- V2/R) (27a)

The above equation is also rewritten as

tann = mK /cos¢ - L/D (27b)

I

where, Ki = (g- V2/R)/D = (gR- V2)/RD (27c)

From (24a), we see that for a given angle of attack =, if altitude H, and

velocity V are kept constant, then the drag D and lift i forces are constant.

The mass m always changes due to thrusting. Then the above relation (27b) for

variable mass can be satisfied in any one of the following three ways.

(i) Variable bank angle with constant angle of attack and thrust angle: With

bank control, (22g) and (27a) mean that the thrust T is constant leading to a

constant mass flow rate.

(ii) Variable thrust with constant bank angle: On the other hand, with thrust

control, (24) implies that we need to change both magnitude and angle of the

thrust, in order to keep a constant drag force. Thus for cruise condition,

both thrust magnitude and angle need to be controlled such that Tcos_ is

constant, but Tsin_ changes according to (24b) [see Fig. 3]. This leads to

variable mass flow rate.

(iii) Variable bank angle and variable thrust: Here, we change both bank angle

and thrust magnitude and angle, in order to satisfy the cruise conditions

(24a) and (24b). This also leads to variable mass flow rate.

Obviously, the bank angle control leading to constant thrust (and hence

constant mass flow rate) seems to be the simplest of all for implementation.

However, it will be interesting to see which of the control schemes provides

greatest amount of heading change and thereby incllnation for the same amount

of fuel expenditure.

16
The bank angle control with constant thrust magnitude and angle has been

thoroughly discussed using arc length as independent variable. 8 However, in

our present work, we continue to use time as independent variable. Assuming

the latitude to be small, the cruise motion is described by

d8

dt - Vcos¢/R (28a)

dm

- K /cosn (28b) dt 2

de _ K tan_ (28c)

dt 3

K = D/gI ; K = g/V - V/R (28d)

2 sp 3

The bank angle control is given by

cosc= mK /[tann + L/D] = mK

1 4

(29a)

where,

K = (g - V2/R) - 2(k 2 - 1) (29b)

4 (L + Otann) pRSC (_)[tann + L/O]

D

and k =_V, the ratio of circular speed to cruise speed at R. From (29), we

see that for a given angle of attack, and at constant altitude, speed, and

thrust angle, the bank angle has to be varied as per the mass. That is, as

mass m decreases along the flight, bank angle _ should be increased. Thus, in

increasing the bank angle with the decrease of mass, we are trying to balance

the decrease in the difference between the vehicle's weight and centrifugal

force with the sum of vertical components of lift and thrust. In this control

scheme, both mass and bank angle change, whereas altitude, velocity, angle of

attack, thrust, thrust angle, and mass flow rate are held constant.

The cruise condition (29) reveals that

17
(a) With v = 0°, there is no banking and the cruise conditions can be

maintained only by variable thrusting.

(b) For 0 < v < 90 °, the cruising speed is less than circular speed. The llft

is directed upward. The gravitational force is higher than the centrifugal

force.

(c) v = 900 corresponds to cruise speed being equal to the circular speed, and

all the aerodynamic force (Tsln_ + L) is used for heading change or turning.

The gravitational force is equal to the centrifugal force.

(d) For v > 90 ° , the cruising speed is higher than the circular speed. The

centrifugal force is higher than the gravitational force and hence the llft is

directed downward in order to prevent the vehicle to escape from Earth.

Given the initial values of mass m{, and heading angle Cj, we find the

initial bank angle _ from (29a). Also-, we can solve (28b) directly as

J

m(t) = - (K /cos_)t + m (30)

2 j

Thus, v(t), and ¢(t) are solved until either of the desired final conditions

m , or @ is realized. With a constant thrust angle _, and given initial mass

n n

mj, and heading angle Cj, and final mass mn (or final heading angle Cn ) the

sequence of solution of the aerocruise problem is to solve, first the mass

equation (28b), second the bank angle equation (29a), and finally the heading

angle equation (28c).

In this formulation for aerocruise, we see that the heading angle changes

with respect to bank angle as given by (28c), and bank angle in turn has to

follow the mass as per (29a), and the mass varies independently according to

(30). Hence, there is no optimization of heading angle w.r.t, bank angle

control variable, for a given fuel consumption or of fuel consumption w.r.t.

bank angle control variable, for a given heading change. Alternatively, from

{29) and (30), we can solve for ¢, which is now a function of thrust angle _,

cruising altitude H, and cruising speed V. Then, we can find the optimal value

18
of n, which should be maintained constant throughout the aerocruise to achieve

maximum _.

Optimization of Heading Change w.r.t. Thrust Angle

The optimization problem here is to find an optimum thrust angle which is

to kept constant throughout the cruise mode, in order to maximize the heading

change. We can solve this problem in a variety of ways. Basically, the heading

angle @ can be solved from (28) to (30) in terms of time t, mass m, or bank

angle g. Thus,

@n = Cj + Ks[In(An/Aj) + Bj - Bn] (31a)

where,

Ks - sinn + (L/D)cosn ] (31b)

A 1 = (1 + singi]/cosgi; Bl = sing- 1' i = j, n (31c)

Note that Ai, and B i can also be expressed in terms of mass m using (29), or

in terms of time t with (30). For example, in terms of mass m, (31c) becomes

1 + Jl- (talK4)2]

Ai = m K ; BI = J1 - (re|K4)2 ; 1 = J, n (31d)

1 4

11

In terms of bank angle g, (31) is rewritten as,

@n = @J + sinn + (L/D)cos_ In osg (1 + singj)J + slngj n

(32)

where, bank angle g Is related with thrust angle n as per (29). Now optimizing

with respect to thrust angle _, (i.e., making d_ /dD = O) and assuming the

n n

initial value @j = O, we get

19
cos_[cos_ + [L/D)sln_] [ In _c°sV osJn(l(I ++ sinVn)l sln_j)j + sln_l - sln_rn] = 0

(33)

Using first order approximations in the change of the bank angle,

cos¢ : cos(_ + A_) s cos¢ - (sine)A_

n J , J J

(34a)

sinc = sin(v + Av) _ sin_ +(cosv )Av

n J J J

and, linearizing the logarithm, the above transcendental equation (33)

becomes,

(34b)

[cosn[cosn + 2 ]

+ cos _ (1 + sin_ )A_ = 0 (3S) (L/D)sinn]sin2e'j j J

Assuming that I + sin_ _ O, and A_ _ O, (35) becomes

J

cosn[cosn + (L/D)sinn]sin2_ + cos2_ = 0

J ]

(36a)

From (29a),

A

cos_j tann + L/D ; A = rajK I (36b)

Using (36b) in (36a),

[tan_ + L/D] [(i/D)tan2_ + {(L/D) 2 - ,42 - l}tan_ - L/D] (37)

1!

Again, assuming tann + L/D _ O, we finally get a simplified form as,

(L/D)tan2n + [(L/D) 2 - A2 - l]tann - L/D= 0 (38)

The implication of tann + L/D = 0 is that (a) 90 o < _ < 180 °, or (b) 270 ° < n

< 3600 . The first condition implies that the thrust T will ald the drag force

2O
instead of opposing it) and the second condition shows that the thrust T will

oppose the lift instead of aiding it.

Summarizing, for bank angle control we have

d_ _ K tanv (39a)

dt 3

dm

-- = - K /cosn (39b)

dt 2

The cruise condition is given by

cosv= mE /[tann + L/D] (39c)

i

The sequence of solutions is

(i) Using initial masses m}, and L/D, solve {38} for optimal thrust angle n.

(ii) Using W, mj, and i/D, solve (39c) for v}.

(iii) Using W, and m , solve for new m from (39b).

]

(iv) Using the mass m, and _, solve (39a) or (31) for new @.

(v) Go to (ii), and repeat the steps.

{vi) Integration stops when m reaches the final m in (39b) and the

n

corresponding maximum heading angle _n is obtained from (39a).

Although (39) can be solved for either set of given values of (a) mj and

m. or {b) ¢} and _n, the condition (38) requires mj and mn to determine the

optimal n which is kept constant throughout the aerocruise mode. Hence, given

the fuel consumption {m - m ), we are trying to determine the maximum heading

J

angle change.

21
V. AEROCRUI SE MODE: THRUST CONTROL

Here, we keep the bank angle constant throughout and change the thrust

magnitude and angle in order to achieve the desired heading change and hence

12

the inclination. For the sake of simplicity, we repeat the equations at

cruise,

d8

- Vcos_/R (40a)

dt

dm

- K /cosn (40b)

dt 2

d_ _ K tanv (40c)

dt 3

The cruise conditions are given by

Tcosn = D (41a)

(Tsinn + L)cosv = m(g - V2/R) (41b)

Combining the two conditions,

tann = mK /cosv- L/D (42)

I

For a constant altitude H, speed V, and given angle of attack _ and bank angle

v, as the mass m changes, the thrust angle _ follows (42). At the same time,

the drag force D is to be kept constant as per (41a). Thus, in order to

satisfy both the conditions (41), we need to adjust thrust magnitude T _nd

angle _ in such a way that Tcosw is kept constant, and Tsin_ changes as per

mass m. From (40), we see that the bank angle is kept constant throughout the

cruise mode and hence the rate of change of heading angle is constant, whereas

the mass flow rate is variable. This is in contrast to the bank angle control

discussed in the last section.

22
Given a constant bank angle v, the initial and final conditions m], @j,

m (or @), the sequence of solutions for the cruise flight with thrust

n n

control is first, solve the cruise condition equation (42) for n, second solve

mass rate equation (40b) for m, and finally solve the heading angle equation

(40c) for @. In the next section we try to find the optimal bank angle which

should be kept constant throughout the cruise, to get maximum heading change.

Optimization of Heading Angle w.r.t. Bank Angle

Here, we are interested in finding the optlmum bank angle so that the

heading change is maximized. For this, we first solve (40)-(42) for the

heading angle and then find the stationary value of @ w.r.t.v. Thus,

¢ = (I g/V)sinvln

n sp

(43a)

where,

2

tannl = miK1/cos_- L/D; sec _i = 1 + tan2nl; i = j, n (43b)

K = (I E/V)sin_ (43c)

6 sp

Alternatively, (43a) can be used to flnd the mass m for a given ¢. Use of the

stationary condition leads to

cos2vln "cosnj(1 + sinnn) ] + sln2 _[ (sinn

cosnn(l + sinnj)J

- slnnj) + (L/D)(cosn n - cosnj)] = 0

(44)

Considering only the first order approximations in the thrust angle n

COSn n = COS(hi + An) _ cosnj - (sinnj)An

(45a)

sinn n = sin(nj + AT) _ sinnj + (cosnj)An

(45b)

23
and linearizing the logarithm, the transcendental equation (44) becomes

°s2v + sin2v _ tai_] lj [1 ÷ si

] n_j

]a_ = 0 (46)

Assuming I + sinai _ O; and &n _ O, (46) reduces to

COS2V + sin2v ____ .

I _ tan2_]j j = 0

Us ing

(47)

tannj = A/cosv - L/D

(48)

in (47). we get a simplified quadratic equation in cosv as

(L/D)Acos2v - [(L/D) 2 + A 2 + l]cosv + (L/D)A = 0 (49)

It is interesting to note that the optimal conditions (38) and (49) for bank

control and thrust control respectively, are Interchangeable by the cruise

condition (36b) or (48).

Summarizing, for thrust control we have

d@ _ K tanv (50a)

dt 3

dm

- K Icosn (50b)

dt 2

The cruise conditions are given by

Tcosn = D (50c)

tann = mK /cos_ - L/D (50d)

1

24
The sequence of solutions is

{i) Using initial mass m and L/D, solve (49) for optimal bank angle v.

]

(li) Using _, m], and L/D, solve (HOd) for W].

(lil) Using n], find the thrust T] from (50c).

(iv) Using _], and v, solve (50) for new _.

(v) Using nj, and mj, solve for new m from (SOb) or (43).

(vl) Go to (ii), and repeat the steps.

(vii) Integration stops when m reaches the final m in (5Oh) and the

n

is obtained from (SOa)

corresponding maximum heading angle _

Although (50) can be solved for either set of given values of (a) m] and

mn or (b) @j and _n, the condition (49) requires m] and m, to determine the

optimal v which is kept constant throughout the aerocruise mode. Hence, given

the fuel consumption (m] - mn )' we are trying to determine the maximum heading

angle change.

25
VI. ASCENT MODE

This is just a replica of the descent mode except for the change in the

mass of the vehicle, and the boundary conditions. Thus, we go through the

equations of motion, change the independent variable to flight path angle, and

finally assume that there is no appreciable change in heading angle. Thus,

dH _ 2m expC_H)sin_/PoSC (51a) d_ n L

dV

- C V/C (51b)

d_ D L

The optimal control problem is posed as follows. Given initial conditions (or"

the conditions at the end of the cruise mode), and the conditions at the end

of the ascent mode, find the optimal control law which maximizes the final

iO

velocity. The performance index is given by

J = - v (52)

£

The Hamiltonian for (51) and (52) is

The adOoint equations are

dAH/d_ = -AH2mn_exp(_H)sIn_/PoSCL (54a)

dA /d_ = A C /C (54b)

v vD L

Solving the state (51) and costate (54) equations, we get

VA = V A = constant (55a)

v n vn

26
exp(Bh) = X exp(BH )

H Hn n

(55b)

The boundary conditions for the adjoint variables are

OJl = - 1

Xv(_ = _r) = _I_ = _r

(56a)

XH(_ = _r ) = _-_ _ = _f

From (5S) and (56),

= 0 (56b)

= 0 (S7)

H

With (57), the Hamiltonian (53) reduces to

_. : - _vV[ CDo/C L + KC L] (58)

The optimal control is then given by

OR

- 0

OC

L

leading to

(59)

c = 4c /K = C (60)

LO DO LE

where, C is the llft coefficient for maximum lift-to-drag ratio (L/D)

LE max

= I/2_ With optimal control (60) in (51), and noting that _ = O, we

DO n

solve for the velocity and altitude as

=E

V(_) = V exp(-_/E)

n

(61)

H= In

2Bm ]-I/_ xp(_BHn) + n (cosy - I)

PoSCLo

(62a)

27
For small _, (62a) reduces to

H= in

exp(-/3H ) _ _.2]

-I//3

At the end of the ascent mode, _ = _f and H = Hf. Then (62) becomes

H = in

f

-I//3

(62b)

(63)

Using (19d) and (63) and noting that He = Hf and Hj = Hn, we get

_f = = _e_'mj/mn

(64)

The relation (64) shows that at the end of the atmospheric phase, the vehicle

has to leave the atmosphere with a positive flight path angle higher in

magnitude to that of the entry flight path angle. This is due to the fact that

the mass m at the beginning of the ascent mode (or end of the cruise mode) is

n

less than the mass m at the end of the descent mode (or beginning of the

J

cruise mode).

28
VII. BOOSTANDREORBITPHASE

During the atmospheric flight, the vehicle performs the desired plane

change and dissipates some energy due to atmospheric drag. Therefore, a second

impulse is required to boost the vehicle back to orbital altitude. The vehicle

exits the atmosphere at point F, with a velocity Vf and flight path angle ;f.

The additional impulse AV required at the exit point F for boosting into an

b'

elliptic orbit with apogee radius R and the reorbit impulse AV required to

c ¢

insert the vehicle into a circular orbit at point C, are obtained by using the

principle of conservation of energy and angular momentum at the exit point F,

11

and the circularization point C. Thus, we have,

(vf + aVb)2/2- _/Ra : (Vc - aV)2/2- c _/Rc

(65)

(Vr + AVb)Rac°s_'f = Rc(Vc - AVc) (66)

Solving for AV and AV from the above equations (65) and (66],

b c

AVb = ]2#(1/R a - 1/Rc)/[1 - (Ra/Rc)2cos2_f] - V¢ (67)

= - J2.(1/R - 1/R I'-/cos f - 1]

C C a C

(68)

Finally, the vehicle is in a circular orbit (of radius R ) moving with

c

the velocity V = Vrg/R .

c c

29
VI I I. NUNERI CA1. DATA AND RESULTS

The following set of data is used for a typical orbital maneuverlng

S-8

research vehicle.

Orbital Data

Altitude of HEO, H = I15,000 m

d

Altitude of LEO, H = IIS,O00 m

c

Altitude of atmospheric boundary, H = Ii0,000 m

a

Radius of Earth, R = 6,356,766 m

E

Acceleration due to gravity at sea level, go = 9.80665 m/sec 2

Atmospheric density at sea level, Po = 1.22S kg/m 3

Gravitational constant of Earth, p = 3.986xi04 m3/sec 2

-1

Inverse atmospheric scale height, _ = 1/7280 m

Vehicle Data

Initial mass, m = 4760 kg

]

- m ) = 1810 kg Propellent available for cruise, (m] n

Final mass, m = 2950 kg

n

2

Aerodynamic reference area, S = 11.613 m

Specific fuel impulse, I = 290 sec

sp

The aerodynamic characteristics are described in terms of the angle of attack

O_ as

C = -2.068686996a 3 + 2.943200144_ 2 + 0.080347684a + 0.031320026

L

(69a)

C = 0.267339707_ 3 + 1.814473159_ 2 - 0.389985867_ + 0.068372034

D

(69b)

3O
Fig. 4 shows the variations of the lift and drag coefficients CL, and C p

Fig. 5 shows the variations of the lift-to-drag ratio E, and the

aeropropulsive efficiency g as a function of the angle of attack _. The

p

maximum lift-to-drag ratio of 2.3149 occurs at the angle of attack of 13

degrees.

and

Deorbit Phase

Initially, the vehicle is at a HEO altitude H of 115 km orbiting with a

d

circular velocity V of 7847.97 m/sec. A deorbit impulse AV of 518.99 m/sec d d

puts the vehicle in an elliptic orbit to intersect the atmospheric boundary at

an altitude H of ii0 km. At the atmospheric entry point, the velocity V is

7334.17 m/sec and the flight path angle _ is -0.77 deE.

e

Atmospheric Phase: Deorbit Mode

b

During the descent mode of atmospheric phase, the vehicle descends from

an altitude H of Ii0,000 m at a velocity V of 7334.17 m/sec to a cruise

altitude H of 72,521 m and the cruise velocity V of 7291.17 m/sec according

J J

to relations (18) and (i9). During this time, the vehicle is maintained at an

angle of attack of 13 degrees corresponding to the lift given by (17) for

maximum lift-to-drag ratio. The time solutions of altitude, velocity, and

flight path angle for the descent mode are shown in Fig. 6. The time taken for

descent mode is found to be 494 seconds.

Cruise Mode: Bank Angle Control

The cruise mode is analyzed using bank angle control or thrust control.

With bank angle control, for the same fuel consumption and a given A/D (or

angle of attack _), the optimum thrust angle as obtained from (38) is

represented in Fig. 7. With this optimum thrust angle, the corresponding

heading angle is obtained from (32) and is shown in Fig. 8. It is seen that at

these constant cruising conditions (of altitude of 72621 m, velocity of 7291.7

m/sec, and thrust of 6108.9 Nw), the maximum heading and hence maximum

inclination of 18.56 degrees is achieved with a thrust angle of 44.72 degrees

and at a higher angle of attack of 24 degrees rather than at the angle of

31
attack of 13 degrees corresponding to maximum L/D. s-8 Corresponding to the

maximum inclination of 18.56 degrees, the time solutions for mass, bank angle,

and heading angle are shown in Fig. 9, where the total time taken for the

cruise mode is 830 seconds.

Cruise Mode: Thrust Control

With thrust control, for the same fuel consumption and a given L/D (or

angle of attack _), the optimum bank angle as obtained from (49) is shown in

Fig. I0. With this optimum bank angle, the corresponding heading angle is

obtained from (43) and is shown in Fig. II. It is seen that at these cruising

conditions, the maximum heading angle of 17.7 degrees is achieved with a bank

angle of 51.9 degrees and at a higher angle of attack of 20 degrees rather

than at the angle of attack of 13 degrees corresponding to maximum L/D.

Corresponding to the maximum heading angle of 17.7 degrees, the time solutions

for mass, thrust, thrust angle, and heading angle are shown in Fig. 12, where

the total time taken for cruise mode is 1345 seconds.

The comparison of maximum heading angle as a function of angle of attack

for both the control strategies shown in Fig. 13, indicates the superiority

of bank control over thrust control. It is to be noted that the heading

achieved depends on the type of control used, cruise conditions, and the angle

of attack.

Atmospheric Phase: Ascent Mode

At the end of cruise mode, the vehicle ascends to the atmospheric

boundary with a constant angle of attack of 13 degrees corresponding to

maximum lift-to-drag ratio as given by (60). At the end of the ascent mode,

the exit velocity V£ is 7238.] m/sec, the flight path angle _f as given by

(64) is 0.9781 deg. The time solutions are shown in FiE. 14.

For the atmospheric phase wlth bank angle control, the total solutions

for altitude, velocity, flight path angle, heading angle, and heating rate are

shown in Fig. I_. Similarly, total solutions for thrust control are shown in

s

Fig. 16. The heating rate is computed from

32
-4 1/2 3.08

O = 3.08xi0 Pk Vk Watts/cm2

where, Pk is expressed in kg/km 3, and V k is expressed in km/sec.

Reorbit Phase

(70)

At the end of the atmospheric phase, a boost impulse AF of 380 m/sec is

b

executed to bring the vehicle to its original altitude H of 110 km. At this

C

time, once again a circularizing impulse AV of 247.97 m/sec is imparted to

C

finally put the vehicle in circular orbit.

33
IX. CONCLUDING REMARKS

We have addressed the synergistic plane change problem in connection with

orbital transfer employing aeroassist technology. The mission involved

transfer from high Earth orbit to low Earth orbit with plane change being

performed within the atmosphere. The complete mission consisted of a deorbit

phase, an atmospheric phase, and finally a reorbit phase. The atmospheric

maneuver was composed of an entry mode, a cruise mode, and finally an exit

mode. The descent and ascent modes have been analyzed using flight path angle

as an independent variable for maximizing the cruise and exit velocities with

a constraint on the minimum cruise altitude. During the cruise mode, constant

altitude and velocity were maintained by means of bank angle control with

constant thrust or thrust control wlth constant bank angle. Conditions have

been obtained for maximizing the heading angle. Under given cruising

conditions, the maximum heading angle has been achieved with an angle of

attack higher than that corresponding to the maximum lift-to-drag ratio.

Comparison between the two control strategies has shown the superiority of

bank control over thrust control in terms of the maximum achievable heading

angle.

Acknowledgements

This research work was supported by the grant NAG1-736 from Spacecraft

Control Branch, NASA Langley Research Center, Hampton, Virginia.

34
X. REFERENCES

I. Walberg, G. D., "A Survey of Aeroasslsted Orbital Transfer, " J. Spacecr_d't,

22, pp. 3-18, Jan.-Feb. 198S.

2. Nyland, F. S., "Considerations of applying continuous thrust during

synergistic plane change", AAS 68-121, AAS/AIAA Astrodynamic Specialists

Conference, Jackson, Wyoming, Sept. 1968.

3. Cuadra, E., and Arthur, P. D., "Orbit plane change by external burning

aerocruise", J. Spacecraft & Rockets, 3, 347-352, March 1966.

4. Clauss, J. S. Jr., and Yeatman, R. D., "Effect of heating restraints on

aeroglide and aerocrulse synergistic maneuver performance", J. Spacecraft and

Rockets, 4, 1107-1109, Aug. 1967.

5. Ikawa, H., and Rudiger, T. F., "Synergistic maneuvering of a winged

spacecraft for orbital plane change", J. Spacecraft & Rockets, 19, 513-520,

Nov.-Dec., 1982.

6. Cervisi, R. T., "Analytical solutions of a cruising plane change maneuver",

J. Spacecraft & Rockets, 22, 134-140, Mar.-Aprll 198S.

7. Blissit, J. A. Jr., "An adaptive guidance algorlthm for an aerodynamlcally

assisted orbital plane change maneuver", M.S. Thesis, Draper Labs., Cambridge,

MA, June 1986.

8. Mease, K. D., Lee, J. Y., and Vlnh, N.-X., "Orbital changes during

hypersonic aerocruise", The J. Astro. Sciences, 36, 103-137, Jan.-June 1988.

9. Marec, J. P., Optimal Space Trajectories, Elsevier Scientific Publlshlng

Company, Amsterdam, 1979.

10. Vlnh, N.-X., Optimal Trajectories in Atmospheric Flight, Elsevier

Scientific Publishing Co., Amsterdam, 1981.

3S
II. Belner, L., "A simplified three-dimensional altitude constrained skip

maneuverfor maximumorbit plane change", Israel J. of Technology, 13, 82-88,

May 1975.

12. Bell, R. N., and Hankey, W. L., Jr., "Applications fo aerodynamic llft in

accomplishing orbital plane change", Air Force Systems Command,

Wrlght-Patterson Air Force BAse, Ohio, ASD-TDR-B3,693, Sept. 1963.

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